Chapter 5.5, Problem 16PSB

To determine

To prove: That the triangle $OGK$ is an isosceles triangle.

Explanation of Solution

Given Information:

  $PHJM$ is a rectangle

  $PG=MK$

Formula used:

Opposite sides of a rectangle are parallel and equal

Basic proportionality theorem

Proof:

  $PHJM$ is a rectangle

We know that, the opposite sides of a rectangle are parallel

  $\Rightarrow PM\parallel HJ\mathrm{...}\left(\text{i}\right)$

In $\Delta OGK,$

  $PM\parallel HJ$

  $\Rightarrow PM\parallel GK\mathrm{...}\left(G-H-J-K\right)$

Using Basic proportionality theorem, “if a line drawn parallel to one side of a triangle intersecting other two sides at two distinct points then it divides the other two sides in the ration” we have

  $\frac{OP}{PG}=\frac{OM}{MK}$

Since $PG=MK\mathrm{...}\left(\text{ii}\right)$

  $\begin{array}{l}\Rightarrow \frac{OP}{MK}=\frac{OM}{MK}\\ \Rightarrow OP=OM\mathrm{...}\left(\text{iii}\right)\end{array}$

Now, $OG=OP+PG$

  $\Rightarrow OG=OM+MK$ … From (ii) and (iii)

  $\Rightarrow OG=OK$

Hence, $\Delta OGK$ is an isosceles triangle.